By Eriko Hironaka
This paintings reports abelian branched coverings of gentle advanced projective surfaces from the topological perspective. Geometric information regarding the coverings (such because the first Betti numbers of a gentle version or intersections of embedded curves) is said to topological and combinatorial information regarding the bottom area and department locus. precise cognizance is given to examples within which the bottom house is the advanced projective airplane and the department locus is a configuration of traces.
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Additional resources for Abelian Coverings of the Complex Projective Plane Branched Along Configurations of Real Lines
Ms to be the matrices defined as follows. (1) If row r of M has only two columns j \ and J2 with entry equal to 1 then let Mr be the 1 x (£ -M) matrix with a 1 in the j i and fa columns and zero elsewhere. , j ^ with entries equal to 1, with d > 2, then let M r be the d x (k -hi) matrix with a 1 in the £,ji and f , H w place for £ = 1 , . . , d, if r is the uth row of M containing more than two entries equal to 1. Let M be the matrix obtained by concatenating these matrices. The following proposition can be checked easily from the definitions.
Are, for C H A P T E R III. HIRZEBRUCH COVERING SURFACES In this chapter we apply the previous results specifically to Hirzebruch surfaces and describe techniques that lead to an effective algorithm for computing the first Betti number 61 and lower bound for the Picard number p of Hirzebruch surfaces X associated to configurations of real lines. The algorithm is given in Chapter IV. We define Hirzebruch covering surfaces and give some properties, following [Hirz], in III. 1. In particular, we describe the generators of the inertia and stabilizer subgroups for the branch locus of p and p.
1 to find the intersection matrix / of the curves in p~~l(C). 3, 6i = 6 J - N u l l ( J ) . A. P o i n t / c u r ve incidence matrix for £ . We begin by ordering the curves in C and the points S of intersection on £. , Lk of lines C and exceptional curves Eq for points q £ T. , ps through which more than two lines pass. Order the curves in £ as follows: L i , . . , Eqt. For each point pr E S with only two lines Ljx and Lj2 passing through p r , set dr = 1. There is a single corresponding point pr in Ljx H Lj7.
Abelian Coverings of the Complex Projective Plane Branched Along Configurations of Real Lines by Eriko Hironaka